\\(\ast\\) is *not* connected in the joined system to either \\(\circ\\) or \\(\bullet\\).
Each system has 3 elements and some number of connections. \\(A\\) has 3 elements and 0 connections (each element is unconnected from each other element). \\(B\\) has 3 elements and 1 connection (\\(\bullet\\) and \\(\circ\\) are connected, \\(\ast\\) is unconnected from them). \\(C = A \vee B\\) also has 3 elements, and 1 connection (the same as \\(B\\)).
Another thing to consider is what is changed by the join? That is, given two systems, do the *elements* change, or the *connections* when you join them? (NB: The connections do not have to change, \\(A\vee A = A\\)).
Originally I used "partition", that's because of how it's drawn. I've updated my description to be about elements and connections.